Research

Research

Numerical Methods for PDEs and Applications

My current research is in its early stages and focuses on learning and developing numerical methods for solving complex partial differential equations (PDEs) with applications in porous media, fracture mechanics, and high-performance computing (HPC). This work aims to build foundational skills for advancing computational techniques in modeling physical phenomena in engineering and the physical sciences.

Key Areas of Focus:

  • Discontinuous Galerkin Methods: Studying and implementing methods for solving Biot equations to model coupled flow and deformation in porous media.
  • Phase-Field Fracture Modeling: Exploring how phase-field components can be integrated into numerical frameworks to simulate crack nucleation and propagation in heterogeneous materials.
  • High-Performance Computing: Learning to use parallel computing frameworks like PETSc and MPI to enhance the scalability and efficiency of solvers for large-scale simulations.

Applications (Future Directions):

  • Energy and Environmental Modeling: Aiming to simulate fluid-structure interactions in porous media for energy resource management and environmental sustainability.
  • Fracture Mechanics: Investigating computational frameworks to predict crack behavior under various loading conditions, providing insights into material failure and safety.
  • Astrophysical Simulations: Planning to extend numerical techniques to analyze complex dynamical systems in astrophysical and cosmological contexts.

This research represents a critical step in building expertise in computational mathematics and applied sciences. As I progress, I aim to bridge the gap between foundational learning and innovative contributions to numerical analysis and HPC. I welcome collaborations with researchers interested in computational methods and their applications.

Dynamical Systems and Cosmology

During my master’s studies at San José State University, my research focused on applying dynamical systems theory to cosmological models in general relativity. This work provided a mathematical framework to study the evolution of the universe and analyze the stability of its critical points.

Key Contributions:

  • Lambda Cold Dark Matter (ΛCDM) Model: Analyzed the stability of critical points in the ΛCDM model, shedding light on transitions between radiation-dominated, matter-dominated, and dark energy-dominated phases of the universe.
  • Geometric Insights: Used dynamical systems techniques to explore the relationship between geometry and energy in cosmological equations.
  • Numerical Simulations: Conducted simulations to verify theoretical findings and visualize trajectories of the universe’s evolution.

Impact:

  • Enhanced understanding of the long-term behavior of cosmological systems.
  • Contributed to the field of mathematical cosmology by providing tools for analyzing nonlinear dynamical systems in general relativity.

This research deepened my appreciation for the interplay between mathematics and physics, and it continues to inform my work on complex systems in applied mathematics.